Layered superconductors as negative-refractive-index metamaterials

We analyze the use of layered superconductors as anisotropic metamaterials. Layered superconductors can have a negative refraction index in a wide frequency range for arbitrary incident angles. Indeed, low-Tc (s-wave) superconductors allow to produce artificial heterostructures with low losses for T<<Tc. However, the real part of their in-plane effective permittivity is very large. Moreover, even at low temperatures, layered high-Tc superconductors have a large in-plane normal conductivity, producing large losses (due to d-wave symmetry). Therefore, it is difficult to enhance the evanescent modes in either low-Tc or high-Tc superconductors.

Metamaterials are attracting considerable attention because of their unusual interaction with electromagnetic waves (see e.g., [1]). In particular, metamaterials supporting negative refractive index have the potential for subwavelength resolution [2] and aberration-free imaging. The first proposed negative index metamaterials used subwavelength electric and magnetic structures to achieve simultaneously negative permittivity ε and permeability µ (see, e.g., [3]). However, these "double negative" structures require intricate design and demanding fabrication techniques, are not "very subwavelength", and suffer from spatial dispersion effects. Moreover, the implicit overlapping electric and magnetic resonances (see, e.g., [4]) often leads to resonant losses that, together with material losses, lead to significant degradation in metamaterial functionality. This manifestation of loss can be quantified by examining the figure of merit (FOM) in such materials, which is defined as |n ′ |/n ′′ where n ′ and n ′′ are the real and imaginary parts of the refractive index n, respectively. The FOMs of negative index materials in the visible and near-IR have experimentally ranged from 0.1 up to 3.5 [3,5].
Another promising route to creating negative index metamaterials is to use strongly anisotropic materials, in particular, uniaxial anisotropic materials with different signs of the permittivity tensor components along, ε , and transverse, ε ⊥ , to the surface (see, e.g., [6,7]).
These materials have been theoretically [8] and experimentally [9] demonstrated to support sub-wavelength imaging, and they have also been proposed as a model system for scatteringfree plasmonic optics [10] and subwavelength-scale waveguiding [11]. These materials are particularly attractive because: they are relatively straightforward to fabricate, compared to double negative metamaterials; they do not require negative permeability; and do not suffer from magnetic resonance losses. The FOMs for such materials have been calculated to be significantly greater than those measured in double negative materials [6,12].
Experimental schemes for creating strongly anisotropic uniaxial materials have typically involved the fabrication of subwavelength stacks of materials whose layers comprise alternating signs of permittivity. For example, alternating stacks of Ag and Al 2 O 3 [9] and of doped and undoped semiconductors [12] have been demonstrated to support strong anisotropy in the visible and infrared frequency ranges respectively. However, spatial dispersion can strongly modify the optical response of the system relative to the ideal effective medium limit response [13]; strong local field variations exist due to the structure and length scales of plasmonic modes supported by negative-permittivity films, even in the limit of l ≪ a 0 , where l is the length scale of the thin films in the material and a 0 is the free-space electromagnetic wavelength. This imposes limitations to subwavelength imaging and waveguiding in such materials. Spatial dispersion may be reduced by making composite structures with thinner layers. However, there exist practical material deposition limitations to thin-film stacks involving film roughness and continuity. In addition, damping due to electron scattering at the thin film interface becomes significant starting at length scales of a 0 v F /c ∼ a 0 /100 where v F is the Fermi velocity in the material [14], limiting the minimum film thickness. It is clear that composite structures are limited in practice as "ideal" strongly anisotropic materials.
We analyze here the idea of using superconductors as metamaterials (see, e.g., [15,16,17]). In particular, we consider layered cuprate superconductors [17] and artificial superconducting-insulator systems [18] as candidates for strongly anisotropic metamaterials. Unlike the composite structures discussed earlier, layered superconductors are not limited in performance by the spatial dispersion effects discussed in [13]. We will analyze these materials in the specific context of subwavelength resolution, which can be achieved by the amplification of evanescent waves [2]. This amplification is high when n is close to unity and its imaginary part is small [2,19]. For the incident p-polarized waves considered here, subwavelength resolution requires Im(ε) ≪ exp(−2k ⊥ L), where k ⊥ is the wavevector component across the surface, and L is the plane lens-thickness [19]. For evanescent modes with k ⊥ = 2ω/c = 2k 0 and L/a 0 = 0.1, we have Im(ε) ≪ 0.081.
We show that in the case of natural high-T c cuprates the losses are high at any reasonable frequency. In the case of artificial layered structures prepared from low-T c superconductors, the losses can be reduced significantly at low temperatures, T ≪ T c , where T c is the critical temperature. The frequency range for such a metamaterial ishω < 2∆, where ∆ is the superconducting gap, which corresponds to a maximum frequency in the THz range for low-T c superconductors. We prove that the in-plane permittivity for low-T c multi-layers is large, preventing the effective enhancement of evanescent waves. This is problematic because subwavelength resolution [2] requires the amplification of evanescent waves. Note that Refs. 16 only focus on the zero-frequency DC case.
Effective permittivity.-We study a medium consisting of a periodic stack of superconducting layers of thickness s and insulating layers of thickness d with Josephson coupling between successive superconducting planes. The number of layers is large, L/(s + d) = N ≫ 1 and s is smaller than: the in-plane magnetic field penetration depth λ , transverse skin depth δ ⊥ (ω), and wavelength a(ω) ∼ 2πc/ω |ε(ω)|. We calculate the effective permittivity, ε = (ε , ε ⊥ ), of the layered system in the case of p-wave refraction.
Layered superconductors with Josephson couplings can be described by the Lawrence-Doniach model, where the averaged current components can be expressed as [20] where ϕ n is the gauge-invariant phase difference between the (n + 1)th and nth superconducting layers, p n is the in-plane superconducting momentum, J c = cΦ 0 /(8π 2 dλ 2 ⊥ ) is the transverse supercurrent density, Φ 0 is the magnetic flux quantum, and λ ⊥ is the transverse magnetic field penetration depths. Also σ ⊥ and σ are the averaged transverse and in-plane quasiparticle conductivities. The transverse E ⊥ and in-plane E components of the electric field are related to the gauge-invariant phase difference and superconducting momentum by [20,21] where is the capacitive coupling between layers, and R D is the Debye length. We linearize the first of Eqs. (1) and consider a linear electromagnetic wave E ,⊥ (x, n, t) = q dk dω (2π) 2 E ,⊥ (k, q, ω) exp(−iωt + ikx + iqn), where q = πl/(N + 1), l = 0, ±1, ±2, and the x-axis is in the plane of the layers. Using Eqs. (1) and (2), we obtain: where ω p = c/(λ ⊥ √ ε) is the Josephson plasma frequency, ε is the interlayer permittivity, γ = λ ⊥ /λ , and q 2 = 2(1 − cos q). Averaged over the sample volume, the Maxwell equation has the form c∇ × H = 4πJ + ∂D/∂t, where D = ε 0 E and In the effective medium approximation, the components of the permittivity tensor can be expressed as [22] ε 0 = (dε + s)/(s + d), and ε 0 ⊥ = ε(s + d)/(sε + d), where we assume that ε superconductor = 1. Fourier transforming the above Maxwell equa- . Therefore, we finally obtain Thus, ε < 0 and ε ⊥ > 0 in the frequency range If the incident angle is close to normal and anisotropy is large, γ ≫ 1, we can find an estimate FOM ≈ 2 Re(ε )/Im(ε ) ≈ εγ 2 ω 3 /2πσ ω 2 p . Electromagnetic waves propagate in the layered superconductors if ω > ω p . Thus, the results obtained are valid if ω p < ω < ω c = 2∆/h. Below we analyze separately the different cases of a typical high-T c layered superconductor, Bi 2 Sr 2 CaCu 2 O 8+δ (Bi2212), and also an artificial low-T c layered structure made from Nb.
Layered high-T c superconductors.-In the case of Bi2212, it is known that s ≪ d = 1-2 nm, ε = 12, α ≈ 0.1, and at low temperatures (T ≪ T c = 90 K) ω p ≈ 10 12 s −1 , γ = 500, σ ≈ 4 · 10 4 Ω −1 cm −1 , and σ ⊥ ≈ 2 · 10 −3 Ω −1 cm −1 [20,23]. In this case, Eqs. (4) can be rewritten as ε ⊥ ≈ ε 1−ω 2 /ω 2 p +4πiσ ⊥ /ω, ε ≈ ε 1−γ 2 ω 2 /ω 2 p +4πiσ /ω. The calculated frequency dependence of the permittivity for Bi2212 is shown in Fig. 1. The superconducting gap for Bi2212 is estimated as ∆ ≈ 2-3k B T c , with ω c ≈ 5 × 10 13 s −1 ≪ γω p . Thus, for any incident angle, Bi2212 has negative n in the frequency range from about 0.15 THz to 7.5 THz, or in the wavelength domain 40 µm < ∼ a < ∼ 2 mm. However, the use of Bi2212 as metamaterial has a disadvantage since the in-plane quasiparticle conductivity σ is large, even at helium temperatures. As it is seen from the inset in Fig. 1 now be rewritten as Therefore, the refraction index n is negative if For artificial structures, γ can be easily made of the order of, or even much larger than, in natural layered superconductors. In contrast to d-wave high-T c superconductors, for bulk s-wave superconductors, the quasiparticle conductivity σ s tends to zero for decreasing T . Thus, in principle, the imaginary part of ε could be made as small as necessary by cooling the system.
Consider now Nb superconducting layers. For estimates we can take [25]: T c = 9. of Nb are well described in the BCS weak-coupling approximation [25]. In particular, its conductivity σ s (ω, T ) can be calculated using the Mattis-Bardeen theory [26] (see inset in Fig. 2). At low temperatures, T ≪ T c , in the weak-coupling BCS limit, we have ∆ = 1.76 k B T c . When ω < ω c and T ≪ T c , we can rewrite the Mattis-Bardeen formula for conductivity [25,26] in the form where t = T /T c . The results of our calculations are shown in Fig. 2. These calculations demonstrate that the losses in artificial structures made from low-T c superconductors can be extremely low. The maximum frequency ω c = 3.52 k B T c /h for Nb corresponds to approximately 0.7 THz. From the results presented in Fig. 2, we can estimate that at ω ∼ ω c the imaginary part of ε is lower than 10 −3 if T < 1 K. At higher frequencies, ω > ω c , the conductivity of the superconductor is about the conductivity of the normal metal and it cannot be easily used as a metamaterial with low losses. Note also that by an appropriate choice of insulator, s, and d, we can vary the parameters γ and ω p in a wide range. If we assume that ε ∼ 10, then to fulfill conditions (7) for ω p < ω c we should prepare highly-anisotropic heterostructures with γ > 10 3 . If the anisotropy is large, we can find from Eq. (6) that Re (ε ) ≈ −c 2 /λ 2 ω 2 . The absolute value of Re (ε ) is very large, |Re (ε )| ≥ c 2 /λ 2 ω 2 c ≈ 3 × 10 6 . These estimates suggest that low-T c superconducting multi-layers might not work as practical metamaterials.
The metamaterial properties of layered superconductors, either natural or artificial, can be tuned varying the temperature or an in-plane magnetic field, which strongly affects the transverse critical current density and, consequently, the plasma frequency. But applying a magnetic field increases dissipation, which is undesirable. Note also that the estimates made above show that the value of FOM may be very large for the systems considered here, however, this does not mean necessarily that these media can be easily used as practical metamaterials.
Cuprates in the normal state.-There is experimental evidence that cuprate superconductors have strongly anisotropic optical characteristics in the normal state [27,28]. For example, it was observed that La 2−x Sr x CuO 4 supports negative permittivity along the CuO planes at frequencies up to the mid-and near-IR range [27]. Moreover, these optical properties could be finely tuned by varying the stoichiometry. Such natural materials are thus candidates for practical anisotropic metamaterials. The use of cuprates in the normal state have evident advantages, such as operating above ω c and to work at room temperature.
However, the normal conductivity of cuprates is of the same order as their quasi-particle conductivity in the superconducting state (see, e.g., the inset in Fig. 1b and Ref. 23). The metamaterial properties of cuprates in the normal state require a separate analysis and will be performed elsewhere.
Conclusions.-Here we analyze the properties of anisotropic metamaterials made from layered superconductors. We show that these materials can have a negative refraction index in a wide frequency range for arbitrary incident angles. However, superconducting metama-